Symplectic homology and Stein manifolds

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AbstractAward: DMS-1005365Principal Investigator: Mark McLeanThe subject area of this project is the symplectic geometry of Stein manifolds. A Stein manifold is a properly embedded complex submanifold of complex affine space. This has a symplectic form induced from the standard one in affine space. If we take a large sphere and intersect it with this Stein manifold then we get another manifold called a Stein fillable contact manifold. Important progress in studying Stein manifolds symplectically was achieved by Eliashberg and Gromov. The primary aim of this project is to find exotic Stein structures and Stein fillable contact structures. The PI intends to prove that there are uncountably many symplectically different Stein structures diffeomorphic to an even dimensional manifold admitting a proper and bounded from below Morse function with only finitely many critical points of index at most half its dimension. The PI also intends to prove that there are infinitely many Stein fillable contact structures on each odd dimensional sphere of dimension 5 and higher, and more generally on Stein fillable contact manifolds obtained from affine varieties. The PI will use an invariant of Stein manifolds called symplectic homology to distinguish these. The PI also aims to show that there is no algorithm to tell you whether one Stein manifold diffeomorphic to affine space is symplectomorphic to another one diffeomorphic to affine space of complex dimension greater than 6. The PI also aims prove a similar undecidability result for contact structures on all odd dimensional spheres of dimension greater than 13. The PI will use an invariant called the growth rate of symplectic homology to achieve this. The PI will use growth rates to show that certain cotangent bundles have many Reeb orbits (even degenerate ones). This generalizes the Gromoll-Meyer theorem. The PI will show that the cotangent bundle of a rationally hyperbolic manifold is not symplectomorphic to a smooth affine variety using growth rates.If we have some classical system such as a pendulum then at any point in time it has a particular position and momentum. If this system has many moving parts such as a double pendulum or a collection free particles then it has many positions and momenta. The set of all such positions and momenta can be encoded in an object called a symplectic manifold. For example the symplectic manifold associated to a pendulum turns out to be a cylinder. Symplectic manifolds are important in many areas of physics such as quantum mechanics and String theory. The PI will study a large class of symplectic manifolds obtained from objects called Stein manifolds. The PI will construct a large list of Stein manifolds called exotic Stein manifolds which look very similar to the symplectic manifold coming from a set of free particles but are actually different if we look at the motion of their respective classical systems. The PI intends to show that there is no computer algorithm telling you if two given exotic Stein manifolds come from the same classical system. This result is useful because it tells us that certain classical systems are very hard to study in general.